On some solutions of Chapman-Kolmogorov equation for discrete-state Markov processes with continuous time

Abstract

The nonlinear equation mentioned in the title is the basic one in the theory of Markov processes. In the case of a discrete-state process, its solution is given by the transition probability function. Usually, solving this equation amounts to solving a linear equation. In 1932, S. N. Bernstein posed the problem of direct determination of the solution. In 1961, such solutions were given in terms of bilinear series by O. V. Sarmanov for stationary continuous-state Markov processes. In 2007, several solutions were obtained by the author in terms of generalized bilinear series without placing Sarmanov’s restrictions. In this paper, our results are extended to discrete-state processes. Two solutions of the Chapman-Kolmogorov equation are derived by means of reducing it to some functional equation. The solutions are represented in the form of a bilinear sum and its generalizations, each term of the sum being proportional to the product of two orthogonal functions. The results obtained are illustrated by two-state processes, which exemplify the assertions derived in this paper. Another example is used to show that the Chapman-Kolmogorov equation has a solution which is devoid of probabilistic sense.